Sharp weighted estimates for multilinear commutators

Sha p weigh ed es ima es o mul ilinea commu a o s C. P´e ez∗and R. T ujillo-Gonz´alez† Jou nal o he London ma hema ical socie y 65 (2002), 672–692. Abs ac We conside mul ilinea commu a o s wi h ec o symbol ~ b= (b1,··, bm) de ined by T~ b( )(x) = ZRn  m Y j=1 (bj(x)−bj(y)) K(x, y) (y)dy, whe e Kis a Calde ´on-Zygmund ke nel. We p o e he ollowing a p io i es ima es o w∈A∞. Fo 0 < p < ∞ he e exis s a cons an Csuch ha kT~ b( )kLp(w)≤Ck~ bk kML(logL)1/ ( )kLp(w) and sup >0 1 Φ(1 )w({y∈Rn:|T~ b (y)|> })≤Csup >0 1 Φ(1 )w({y∈Rn:ML(logL)1/ (||~ b|| )(y)> }) whe e ||~ b|| =Qm j=1 kbjkoscexpL j, Φ( ) = log1/ (e+ ), 1 =1 1+··+1 m, and ML(logL)αis an O licz ype maximal ope a o . This ex ends, wi h a di e en app oach, classical esul s by Coi man [5] (see also [6] [24]). As a co olla y we deduce ha he ope a o s T~ ba e bounded on Lp(w) when w∈Apand ha hey sa is y co esponding weigh ed L(logL)1/ ype es ima es wi h w∈A1. ∗Pa ially suppo ed by DGESIC G an PB980106. †Pa ially suppo ed by Gobie no de Cana ias PI1999/105. 1991 Ma hema ics Subjec Classi ica ion: 42B20, 42B25. Keywo ds: Calde ´on-Zygmund singula in eg al ope a o s, commu a o s, Apweigh s, maximal unc ions 1 1 In oduc ion The main pu pose o his a icle is o p o e sha p es ima es o mul ilinea commu a o s in ol - ing nons anda d symbols. This will be es ablished by means o app op ia e maximal ope a o s which somehow con ol he commu a o s. This illus a es he classical Calde ´on–Zygmund p in- ciple which oughly s a es ha any singula in eg al ope a o is con olled by a sui able maximal ope a o . Backg ound Mo i a ed by he wo k o Calde ´on on commu a o s, Coi man, Rochbe g and Weiss in o- duced in [7] he ope a o Tb (x) = ZRn (b(x)−b(y))K(x, y) (y)dy, (1.1) whe e Kis a ke nel sa is ying he s anda d Calde ´on-Zygmund es ima es (see sec ion 3.1) and whe e b, he “symbol” o he ope a o , is any locally in eg able unc ion. The ope a o is called commu a o since Tb= [b, T] = bT −T(b·) whe e Tis he Calde ´on-Zygmund singula in eg al ope a o associa ed o K. The main esul om [7] s a es ha [b, T] is a bounded ope a o on Lp(Rn), 1 < p < ∞, when he symbol bis a BMO unc ion. These commu a o s ha e p o ed o be o in e es in many si ua ions. We shall only men- ion he ecen esul s in he heo y o non di e gence ellip ic equa ions wi h discon inuous coe icien s [2] [3] [9]. The e is also an in e es ing connec ion, as poin ed ou in [21], wi h he nonlinea commu a o conside ed by R. Rochbe g and G. Weiss in [23] and de ined by →N =T( log | |)−T log |T |. This is in u n is ela ed o he Jacobian mapping o ec o unc ion and wi h nonlinea P.D.E. as shown in [13] [14]. A na u al gene aliza ion o he commu a o [b, T] is gi en by Tm b (x) = ZRn (b(x)−b(y))mK(x, y) (y)dy, (1.2) whe e m∈N. The case m= 0 ecap u es he Calde ´on-Zygmund singula in eg al ope a o . Obse ing ha Tm b= (m imes) z }| { [b, ··,[b, T]] we ha e ha o m∈N,Tm bis also a bounded ope a o on Lp(Rn), 1 < p < ∞, when bis a BMO unc ion. I was shown in [21] ha he e is an in ima e connec ion be ween he commu a o Tm band i e a ions o he maximal ope a o s. Indeed, he main heo em om [7] was sha pened in [21] as ollows: o any 0 < p < ∞and any w∈A∞ he e is a cons an Csuch ha ZRn |Tm b (x)|pw(x)dx ≤Ckbkmp BMO ZRn (Mm+1 (x))pw(x)dx, (1.3) 2 whe e Mm+1 deno es he m+ 1 i e a ions o he Ha dy-Li lewood maximal ope a o , namely Mm= (m imes) z }| { M◦··◦M. Fu he mo e, his inequali y is sha p since, by he Lebesgue di e en ia ion heo em, Mm+1 can no be eplaced by he smalle ope a o Mm. Es ima e (1.3) can also be seen as a gene aliza ion o a by now classical esul o Coi man o Calde ´on-Zygmund singula in eg al ope a o s. See [5] and also [6]. The e is a weak e sion o inequali y (1.3) ob ained in [20]. Indeed, i we le Φm( ) = logm(e+ ), o any w∈A∞and b∈BMO he e is a cons an Csuch ha sup >0 1 Φm(1 )w({y∈Rn:|Tm b (y)|> })≤Csup >0 1 Φm(1 )w({y∈Rn:Mm+1 (y)> }).(1.4) A p io i inequali ies o he o m (1.3) (o (1.4)) encode a good amoun o in o ma ion abou he beha io o he ope a o . The i s obse a ion is ha in any case hey e lec he highe deg ee o singula i y o Tm b, as compa ed wi h T, since a la ge ope a o han M, namely Mm+1, is needed o balance he inequali y. As a second ins ance, i p > 1 we can apply m+ 1 imes Muckenhoup ’s heo em o conclude om (1.3) he well known ac ha high o de commu a o s a e bounded on Lp(w) whene e w∈Ap. I should be men ioned ha his Apes ima e ollows, as i is well known, om an es ima e due o S ¨ombe g (see [15, p. 268], [25, p. 417]). This me hod is powe ul and can be applied o di e en si ua ions. As a sample we emi o [10] o a e y nice applica ion o commu a o s wi h s ongly singula in eg al ope a o s. Howe e , his es ima e o S ¨ombe g is no sha p enough nei he o de i e (1.3) no (1.4). In ac i can be shown om (1.4) he ollowing (Llog L ype) endpoin es ima e i s deduced in [20]: le w∈A1and b∈BMO, hen he e exis s a cons an Csuch ha o all λ > 0 w({y∈Rn:|Tm b (y)|> λ})≤CZRn | (y)| λ(1 + log+(| (y)| λ))mw(y)dy. (1.5) As a hi d ins ance, i was shown in [21] ha es ima e (1.3) implies a sha p wo weigh ed inequali ies o he commu a o o he o m ZRn |Tm b (x)|pw(x)dx ≤Ckbkmp BMO ZRn | (x)|pM[(m+1)p]+1w(x)dx, whe e no condi ion on he weigh wis assumed. This esul is an ex ension o he case m= 0 p o ed in [18] gene alizing some p e ious pa ial esul s by M. Wilson [26]. The app oach conside ed in [18] is di e en om ha in [26] and i combines (1.3) wi h ce ain sha p wo weigh ed es ima es o he Ha dy–Li lewood maximal unc ion de i ed in [19]. Finally, he e is a ela ionship be ween (1.4) and he endpoin beha io o he ope a o . Indeed, an in e es ing obse a ion ha occu s in he de elopmen o he heo y o commu a o s is ha he Lp heo y, p > 1, was de eloped wi hou appealing o any endpoin es ima e. On he o he hand, i is well known ha in he classical Calde ´on-Zygmund Lp heo y, p > 1, a c ucial s ep is o show ha he ope a o s a e o weak ype (1,1). Howe e , his is no he case o he commu a o [b, T] when b∈BMO as shown in [20] and es ima e (1.5) is he igh eplacemen . 3 This can be seen as ano he way o exp essing he ac ha commu a o s ha e a highe deg ee o singula i y as compa ed wi h he Calde ´on-Zygmund singula in eg al ope a o s. Resul s o his pape In his pape we ob ain simila es ima es o a wide class o commu a o s. Gi en a Calde ´on- Zygmund singula in eg al ope a o Twi h ke nel Kin Rn×Rnand a ec o ~ b= (b1, .., bm) o locally in eg able unc ions, we de ine he mul ilinea ope a o T~ b (x) = ZRn  m Y j=1 (bj(x)−bj(y)) K(x, y) (y)dy, (1.6) gene alizing he commu a o (1.2). We emi he eade o [6] o an ex ensi e s udy o mul i- linea ope a o s. We will be conside ing he ollowing class o symbols: o ≥1 and o any locally in eg able unc ion , we de ine k koscexpL = sup Q k − QkexpL ,Q being he sup emum aken o e all he cubes Qwi h sides pa allels o he axes. He e kgkexpL ,Q is he mean alue o gon he cube Qwi h espec o he Young unc ion Φ( ) = e −1. See Sec ion 2.2 o he p ecise de ini ion. As usual, Qdeno es he a e age o on Q. Fo ≥1 we de ine he space OscexpL by OscexpL ={ ∈L1 loc(Rn) : k koscexpL <∞}. In he pa icula case o = 1, OscexpL1coincides wi h he BMO space by John-Ni embe g’s heo em. Also we ha e OscexpL BMO o any > 1. O he examples a e p o ided by T udinge ’s inequali y o Riesz po en ials. To be mo e p ecise, o any 0 < α < n and any ∈Ln/α(Rn), he Riesz po en ial Iα o o de αbelongs o OscexpL(n/α)0(c . [12, 27]) Fo he s a emen o ou esul s we in oduce some no a ion ha will simpli y he p esen- a ion. Along his pape mwill always be he numbe o symbols o he ope a o T~ bwhe e ~ b= (b1, .., bm) is a amily o mlocally in eg able unc ions. I 1, .., ma e mposi i e eal numbe s, we deno e 1 =1 1 +··+1 m and ||~ b|| = m Y j=1 kbjkoscexpL j. Ou main esul s a e he ollowing. Theo em 1.1 Le 0< p < ∞,w∈A∞. Suppose ha T~ b is he commu a o (1.6) whe e bis as abo e such ha bi∈OscexpL i, i≥1,1≤i≤m. Then he e exis s a cons an C > 0such ha ZRn |T~ b( )(x)|pw(x)dx ≤C||~ b||pZRn (ML(logL)1/ ( )(x))pw(x)dx (1.7) 4 o all bounded unc ions wi h compac suppo . Fo he p ecise de ini ion o he ope a o ML(logL)αsee Sec ion 2.2. Inequali y (1.7) shows ha he maximal ope a o ML(logL)1/ is he one ha con ols he mul ilinea commu a o s Tb. Since i≥1 o each i, i ollows ha ML(logL)1/ is poin wise smalle han ML(logL)mand his in u n is known o be equi alen o m+ 1 i e a ions o he Ha dy–Li lewood maximal ope a o Mm+1. Hence we can use again Muckenhoup ’s heo em and deduce he boundedness o Tbon Lp(w) o p > 1 and any w∈Ap: Co olla y 1.2 Le 1<p<∞and w∈Ap. Suppose ha T~ b is he commu a o (1.6) whe e bis as abo e such ha bi∈OscexpL i, i≥1,1≤i≤m. Then he e exis s a cons an C > 0 such ha ZRn |T~ b( )(x)|pw(x)dx ≤C||~ b||pZRn | (x)|pw(x)dx (1.8) o all bounded unc ions wi h compac suppo . As we men ioned abo e his es ima e is a gene alized e sion o Coi man’s esul in [5]. Howe e , ou app oach is di e en and will be based on a poin wise inequali y ha oughly can be exp essed as M# δ(T~ b )(x)≤C||~ b||ML(logL)1/ (x) + R( )(x) (1.9) o an app op ia e posi i e bu small enough numbe δ.Rdeno es ce ain “ emainde ” ope a o which is smoo he , less singula , han ML(logL)1/ in some sui able sense. See Lemma 3.1 o a comple e and p ecise s a emen o his es ima e. Mo eo e , easoning as in [21], Theo em 2, om his heo em we deduce he ollowing es ima e o gene al weigh s. As usual [α] will deno e he in ege pa o α. Theo em 1.3 Le 1< p < ∞and le T~ bbe as in Theo em 1.1. Then he e exis s a cons an C such ha o any weigh w ZRn |T~ b( )(x)|pw(x)dx ≤C||~ b||pZRn | (x)|pM[( 1 +1)p]+1(w)(x)dx o all bounded unc ions wi h compac suppo . Rema k 1.4 We ema k ha he numbe o i e a ions o he maximal unc ion needed in he Theo em is op imal as can be seen in [21], Sec ion 5. In ac i ollows om he p oo o Theo em 1.3 ha he e is a sha pe es ima e: ZRn |T~ b( )(x)|pw(x)dx ≤C||~ b||pZRn | (x)|pML(log L)(1 +1)p−1+(w)(x)dx whe e  > 0, being he esul alse o = 0. 5 The key poin wise es ima e (1.9) is also used o de i e he nex endpoin esul . Recall ha commu a o s wi h BMO unc ions a e no o weak ype (1,1). Recall ha we deno e ||~ b|| =Qm j=1 kbjkoscexpL jand Φ( ) = log1/ (e+ ), whe e 1 = 1 1+··+1 m. Theo em 1.5 Le w∈A1. The e exis s a cons an C > 0such ha o all λ > 0 w({y∈Rn:|Tb (y)|> λ})≤CZRn Φ(||~ b|| | (y)| λ)w(y)dy, (1.10) o all bounded unc ions wi h compac suppo and o all ~ b. The p oo is based on he ollowing esul which gene alizes inequali y (1.4). Theo em 1.6 Le w∈A∞. Then, he e exis s a posi i e cons an Csuch ha sup >0 1 Φ(1 )w({y∈Rn:|Tb (y)|> })≤Csup >0 1 Φ(1 )w({y∈Rn:MΦ(||~ b|| )(y)> }) (1.11) o all bounded unc ions wi h compac suppo . In any o he abo e esul s, i we choose b1=·· =bmand 1=·· = m= 1, we eco e comple ely he co esponding esul s om [20] and [21]. As usual Cwill deno e a posi i e cons an ha can change i s alue on each s a emen . I he cons an depends on p ecise pa ame e s i will be poin ed ou . 2 P elimina ies In his sec ion we in oduce he basic ools needed o he p oo o he main esul s. 2.1 The Fe e man-S ein inequali y o A∞weigh s A weigh will always mean a posi i e unc ion which is locally in eg able. We say ha a weigh wbelongs o he class Ap, 1 < p < ∞, i he e is a cons an Csuch ha 1 |Q|ZQ w(y)dy 1 |Q|ZQ w(y)1−p0dyp−1 ≤C o each cube Qand whe e as usual 1 p+1 p0= 1. A weigh wbelongs o he class A1i he e is a cons an Csuch ha 1 |Q|ZQ w(y)dy ≤Cin Qw. We will deno e he in imum o he cons an s Cby [w]Ap. Obse e ha [w]Ap≥1 by Jensen’s inequali y. 6 Since he Apclasses a e inc easing wi h espec o p, he A∞class o weigh s is de ined in a na u al way by A∞=∪p>1Ap. Howe e , he ollowing cha ac e iza ion i is mo e in e es ing: he e a e posi i e cons an s cand ρsuch ha o any cube Qand any measu able se Econ ained in Q hen w(E) w(Q)≤c|E| |Q|ρ . A simple ac ha will be use ul is ha , o any w∈Apand any m > 0, wm= min{w, m} ∈ Apwi h [wm]Ap≤Cp[w]Ap. Fo mo e in o ma ion on Apweigh s we emi he eade o he e e ences [11] and [25]. We ecall now he de ini ions o classical maximal ope a o s. I , as usual, Mdeno es he Ha dy–Li lewood maximal ope a o , we conside o δ > 0 Mδ (x) = M(| |δ)1/δ(x) = sup Q3x 1 |Q|ZQ | (y)|δdy!1/δ , M#( )(x) = sup Q3x in c 1 |Q|ZQ | (y)−c|dy ≈sup Q3x 1 |Q|ZQ | (y)− Q|dy and a a ian o his sha p maximal ope a o ha will become he main ool in ou scheme M# δ (x) = M#(| |δ)(x)1/δ. The main inequali y be ween hese ope a o s o be used is a e sion o he classical one due o C. Fe e man and E. S ein (see [24], [16]). Lemma 2.1 Le wbe an A∞weigh , hen he e exis s a cons an cdepending upon he A∞ condi ion o wsuch ha o all λ,  > 0 w({y∈Rn:M (y)> λ, M# (y)≤λ})≤c ρw({y∈Rn:M (y)>λ 2}). As a consequence we ha e he ollowing es ima es o δ > 0. a) Le ϕ: (0,∞)→(0,∞)doubling. Then, he e exis s a cons an cdepending upon he A∞ condi ion o wand he doubling condi ion o ϕsuch ha sup λ>0 ϕ(λ)w({y∈Rn:Mδ (y)> λ})≤csup λ>0 ϕ(λ)w({y∈Rn:M# δ (y)> λ}) (2.1) o e e y unc ion such ha he le hand side is ini e. b) Le 0<p<∞ he e exis s a posi i e cons an Cupon he A∞condi ion o wand psuch ha ZRn (Mδ (x))pw(x)dx ≤CZRn (M# δ (x))pw(x)dx, (2.2) o e e y unc ion such ha he le hand side is ini e. 7 2.2 O licz maximal unc ions By a Young unc ion Φ we shall mean a con inuous, nonnega i e, s ic ly inc easing and con ex unc ion on [0,∞) wi h lim →0+ Φ( ) = lim →∞ Φ( )= 0. We de ine he Φ-a e ages o a unc ion o e a cube Qby k kΦ,Q =k kΦ(L),Q = in {λ > 0 : 1 |Q|ZQ Φ| (x)| λdx ≤1}. Fo O licz no ms we a e usually only conce ned abou he beha io o Young unc ions o la ge alues o ’s. Gi en wo unc ions Band C, we w i e B( )≈C( ) i B( )/C( ) is bounded and bounded below o ≥c > 0. We also ecall ha i B( )≤C( ) o ≥c > 0, hen k kB,Q ≤Ck kC,Q wi h Can absolu e cons an . Fo mo e in o ma ion on he subjec see he e e ence [22]. Associa e o his a e age we can de ine a maximal ope a o MΦgi en by MΦ (x) = MΦ(L) (x) = sup Q3x k kΦ,Q, whe e he sup emum is aken o e all he cubes con aining x. Fo example, i Φ(x) = ex −1, hen k·kexpL ,Q and MexpL , deno e espec i ely he Φ-a e age and he maximal ope a o associa ed o Φ. Simila ly we ha e o Φ(x) = xlog (e+x) k · kL(logL) ,Q and ML(logL) . Obse e ha by he abo e ema ks, M ≤CML(logL) o any > 0. These examples will be ele an in ou wo k. Finally, we will be using he ollowing known poin wise inequali y: i m∈N hen ML(logL)m∼Mm+1 = (m+1 imes) z }| { M◦··◦M, he m+ 1 i e a ions o he Ha dy-Li lewood maximal ope a o . A gene aliza ion o his equi - alence can be ound in [8]. We begin wi h some echnical lemmas on con ex unc ions whose p oo s a e s anda d. Lemma 2.2 ([17], Lema 2.1) I Φ0,Φ1, ..., Φma e eal- alued, non-nega i e, nondec easing, le con inuous unc ions de ined on [0,∞)such ha , wi h he de ini ion Φ−1 i(x) = in {y: Φi(y)> x}, i e i y Φ−1 1(x)Φ−1 2(x)· ·Φ−1 m(x)≤Φ−1 0(x),(2.3) 8 hen o all 0≤x1, x2, .., xm<∞ Φ0(x1x2· ·xm)≤Φ1(x1)+Φ2(x2) + ··+Φm(xm) Lemma 2.3 Le Φ0be a con ex unc ion such ha Φ0(0) = 0 and le Φ1, .., Φmas in he p e ious lemma all e i ying (2.3). I 1, 2, .., ma e unc ions sa is ying ha k ikΦi,Q <∞ o all 1≤i≤mand o a gi en cube Q, hen k 1··· mkΦ0,Q ≤mk 1kΦ1,Q ···k mkΦm,Q.(2.4) PROOF: Taking ε > 0 small enough, by Lemma 2.2 and con exi y 1 |Q|ZQ Φ0 1(x) 2(x)· · m(x) m(k 1kΦ1,Q +ε)(k 2kΦ2,Q +ε)· ·(k mkΦm,Q +ε)dx ≤ 1 m 1 |Q|ZQ Φ0 1(x) 2(x)· · m(x) (k 1kΦ1,Q +ε)(k 2kΦ2,Q +ε)· ·(k mkΦm,Q +ε)dx ≤ 1 m 1 |Q|ZQΦ1 1(x) k 1kΦ1,Q +ε+··+Φm 1(x) k mkΦm,Q +εdx ≤1 which implies ha k 1 2· · mkΦ0,Q ≤m(k 1kΦ1,Q +ε)· ·(k mkΦm,Q +ε). Now, aking le ing ε→0 we ge (2.4). The main example ha we will be using is he ollowing: 1 |Q|ZQ | 1··· mg| ≤ Ck 1kexpL 1,Q ···k mkexpL m,QkgkL(logL)1/ ,Q,(2.5) whe e 1, .., m≥1 and 1 =1 1 +··+1 m . Indeed, in his case we can w i e o any x≥0 ha log 1 1(1 + x)···log 1 m(1 + x)x log 1 1+··+1 m(e+x) ≤x. Then (2.3) holds o Φ0(x) = x, Φ−1 i(x) = log 1 i(1 + x), i= 1, .., m and Φ−1 m+1(x) = x log 1 1+··+1 m(e+x) . The in e se unc ions a e gi en by Φi(x) = ex i−1, i= 1, .., m, and Φm+1(x)≈xlog 1 1+··+1 m(e+ x). 9 Now, as in (3.4) and making use again o Kolmogo o ’s inequali y and (2.5), we can es ima e IV by IV ≤C |2B|Z2B |b1(y)−λ1|···|bm(y)−λm|| (y)|dy ≤Ckb1−λ1kexpL 1,2B···kbm−λmkexpL m,2Bk kL(logL)1/ ,2B ≤CML(logL)1/ (x),(3.10) whe e ecall ha 1 =1 1 +··+1 m . Finally, o V, choosing c= (T((b1−λ1)··(bm−λm) 2)))Band epea ing he a gumen used o ge (3.5), i ollows om (2.5) and (3.6) ha V≤C ∞ X k=1 2−kγ 1 (2k+1R)nZ2k+1B  m Y j=1 |bj(w)−λj| | (w)|dw ≤C ∞ X k=1 2−kγ   m Y j=1 kbj−λjkexpL j,2k+1B k kL(logL)1/ ,2k+1B ≤C"∞ X k=1 2−kγkm#  m Y j=1 kbjkoscexpL j ML(logL)1/ (x).(3.11) Finally, om (3.10) and (3.11) we conclude III ≤CML(logL)1/ (x), which oge he wi h (3.8) and (3.9) gi es (3.1) and he p oo o he lemma is inished. We a e now in a posi ion o p o e Theo em 1.1 P oo o Theo em 1.1: We may assume ha ZRn (ML(logL)1/ (x))pw(x)dx < ∞,(3.12) since o he wise he e is no hing o be p o ed. To apply he Fe e man-S ein inequali y (2.2) we i s ake o g an ed ha kMδ(T~ b )kLp(w) is ini e. We will check his o he end o he p oo . 16 We p oceed by induc ion on m. Fo m= 1, by (2.2) and Lemma 3.1 we can es ima e kTb1 kLp(w)≤ kMδ(Tb1 )kLp(w) ≤CkM# δ(Tb1 )kLp(w) ≤Ckb1koscexpL 1hkMε(T )kLp(w)+kML(logL)1/ 1 kLp(w))i ≤Ckb1koscexpL 1hkT kLp(w)+kML(logL)1/ 1 kLp(w)i ≤Ckb1koscexpL 1hkM kLp(w)+kML(logL)1/ 1 kLp(w)i ≤Ckb1koscexpL 1kML(logL)1/ 1 kLp(w) whe e he ou h inequali y ollows since w∈A∞and he e o e he e exis s q > 1 such ha w∈Aq, hen we can choose εsuch ha 0 < ε < p/q (we may ake q > p i necessa y). The i h holds by he classical es ima e (1.3) o m= 0 (see [6], Chap e 1). Suppose now ha o m−1 he heo em is ue and le ’s p o e i o m. So, he same a gumen used abo e and by he induc ion hypo hesis gi es kT~ b kLp(w)≤ kMδ(T~ b )kLp(w) ≤CkM# δ(T~ b )kLp(w) ≤Chkb1koscexpL 1· ·kbmkoscexpL mkML(logL)1/ kLp(w) + m−1 X j=1 X σ∈Cm j kbσkoscexpLσkMε(T~ bσ0)kLp(w)  ≤Chkb1koscexpL 1· ·kbmkoscexpL mkML(logL)1/ kLp(w) + m−1 X j=1 X σ∈Cm j kbσkoscexpLσkbσ0koscexpLσ0kML(logL)1/ σ0 kLp(w)  ≤Ckb1koscexpL 1· ·kbmkoscexpL mkML(logL)1/ kLp(w), since ML(logL)1/ σ0≤C ML(logL)1/ Le us check now ha o app op ia e δwe ha e kMδ(T~ b )kLp(w)<∞. Indeed, as abo e since w∈A∞, he e exis s q > 1 such ha w∈Aqand we can choose δsmall enough so ha p/δ > q. Then by Muckehoup ’s heo em all is educed o checking ha kTb kLp(w)<∞. Suppose ha he symbols bk’s and he weigh wa e all bounded unc ions. Since has compac suppo we may assume ha he suppo o is con ained in he ball BR=B(0, R). Then we can spli he in eg al as ZRn |T~ b (x)|pw(x)dx =Z|x|≤2R |T~ b (x)|pw(x)dx +Z|x|>2R |T~ b (x)|pw(x)dx. 17 The i s in eg al can be easily es ima ed making use o he L∞–boundedness o he bk’s and wand he Lq–boundedness o q > 1 o he Calde ´on–Zygmund ope a o T. Fo he second e m, by he p ope ies o he ke nel Kand he boundedness o he symbols bk’s, since |x|>2R, we ha e he ollowing poin wise es ima e |T~ b (x)| ≤ CZBR |b1(x)−b1(y)|···|bm(x)−bm(y)|| (y)| |x−y|ndy ≤C |x|nZB(0,|x|) | (y)|dy ≤CM (x) ≤CML(logL)1 (x).(3.13) Thus, Z|x|>2R |T~ b (x)|pw(x)dx ≤CZ|x|>2R (ML(logL)1 (x))pw(x)dx which is ini e by he assump ion (3.12). Fo he gene al case, we will unca e he symbols bk’s and he weigh was ollows (c [6], p. 40). We deno e by ~ bN he ec o o unca ed elemen s by N, i.e., ~ bN= (bN 1, .., bN m) whe e each bN kis he unca ion o bkas i is de ined in (2.6). Obse e ha in ou case (2.7) becomes kbN kkoscexpL k≤CkbkkoscexpL k(3.14) wi h C > 0 a cons an independen o N. Analogously we conside he unca ions o he weigh wby wN= in {w, N} ha sa is y [wN]A∞≤C[w]A∞.(3.15) Then (1.7) holds o he ope a o T~ bNand he weigh wN. Combining (3.14) and (3.15), his es ima e gi es ZRn |T~ bN (x)|pwN(x)dx ≤C  m Y j=1 kbjkp oscexpL j ZRn (ML(logL)1/ (x))pw(x)dx. Nex , aking in o accoun ha has compac suppo , we deduce ha any p oduc bN i1··bN ik con e ges in any Lq o q > 1 o bi1· ·bik as N→ ∞. Hence, he classical Lq–boundedness o he ope a o Tgi es, a leas o a subsequence, ha |T~ bN (x)|pwN(x) con e ges poin wise almos e e ywhe e o |T~ b (x)|pw(x) and by Fa ou’s lemma we conclude he heo em o his gene al case. The heo em is p o ed. 3.2 P oo o Theo em 1.5 We adap he e some o he a gumen s om [20]. Since he p oo o Theo em 1.5 is based on Theo em 1.6 we p o e his i s , namely we mus show ha sup >0 1 Φ(1 )w({y∈Rn:|T~ b (y)|> })≤Csup >0 1 Φ(1 )w({y∈Rn:MΦ(||~ b|| )(y)> }) (3.16) 18 o all bounded unc ions wi h compac suppo . Recall ha Φ( ) = Φ~ b( ) = log1/ (e+ ). In ac we a e going o p o e some hing s onge han (3.16) namely: Fo e e y ~ b,ϕ: (0,∞)→(0,∞) doubling wi h ϕ( )≤C , > 0, and o e e y 0< δ < 1 he e exis s a cons an Csuch ha sup >0 ϕ( )w({y∈Rn:Mδ(T~ b )(y)> })≤Csup >0 ϕ( )w({y∈Rn:MΦ(||~ b|| )(y)> }) (3.17) o all bounded unc ions wi h compac suppo . By he Lebesgue di e en ia ion heo em and aking ϕ( ) = Φ(1 )−1= log1/ (e+1 )i is clea ha (3.17) implies (3.16). By making use o he weigh ed e sion o he Fe e man-S ein lemma 2.1, mo e p ecisely es ima e (2.1), we ha e ha sup >0 ϕ( )w({y∈Rn:Mδ(T~ b )(y)> })≤Csup >0 ϕ( )w({y∈Rn:M# δ(T~ b )(y)> }) (3.18) whene e he le hand side is ini e. The e o e (3.17) will ollow om sup >0 ϕ( )w({y∈Rn:M# δ(T~ b )(y)> })≤Csup >0 ϕ( )w({y∈Rn:MΦ(||~ b|| )(y)> }).(3.19) We i s check ha he le hand side o (3.18) is ini e o all bounded unc ion wi h compac suppo . By p oceeding as in he p oo o Theo em 1.1, we may assume ha band w a e bounded. Fo he gene al case o unbounded symbols and unbounded weigh we ep oduce he a gumen used in he p oo o Theo em 1.1 aking in o accoun his ime he weak–(1,1) boundedness o he ope a o Twhich gi es he con e gence in measu e. Suppose ha supp ⊂BR=B(0, R). Hence, since 0 < δ < 1, i ollows ϕ( )w({y∈Rn:Mδ(T~ b )(y)> })≤C ϕ( )|{y∈Rn:Mδ(χB2RT~ b )(y)> /2}| +C ϕ( )|{y∈Rn:Mδ(χRn B2RT~ b )(y)> /2}| =I+II. Fo Iwe use ha Mis o weak ype–(1,1) and he ac ha ϕ( )≤C , hen I≤C |{y∈Rn:M(χB2RT~ b )(y)> /2}| ≤CZB2R |T~ b (y)|dy ≤CRn/2ZRn |T (y)|2dy1/2 , which is ini e since Tis a Calde ´on–Zygmund ope a o and using ha he symbols bk’s a e bounded. 19 Fo II we ake in o accoun he poin wise es ima e (3.13) and he well known ac ha (M )δ∈A1, hen we ha e II ≤C |{y∈Rn:Mδ(M )(y)> C }| ≤C |{y∈Rn:M (y)> C }| ≤CZRn | (y)|dy < ∞. Combining he homogenei y and he linea i y o T~ b, i is easy o see ha we may assume ha ||~ b|| = 1 in bo h (1.10) and (3.16). To p o e (3.19) we p oceed by induc ion on m. 3.3 The case m= 1 This case is essen ially aken om [20] and we epea i , wi h mino modi ica ions, o he sake o comple eness. In his case he ope a o Tbis simply de ined by one single unc ion b Tb = [b, T] =b T( )−T(b ), whe e Tis any Calde ´on–Zygmund ope a o . Recall ha by homogenei y we may assume ha ||b|| =kbkoscexpL = 1 and he e o e wha we mus p o e is sup >0 ϕ( )w({y∈Rn:M# δ([b, T] )(y)> })≤Csup >0 ϕ( )w({y∈Rn:ML(log L)1/ ( )(y)> }) (3.20) o all bounded unc ions wi h compac suppo . Now applying Lemma 3.1 wi h any αsuch ha δ < α < 1 we ha e ha he le hand side o (3.20) is es ima ed by Csup >0 ϕ( )w({y∈Rn:ChML(log L)1/ ( )(y) + Mα(T )(y)i> }) ≤Csup >0 ϕ( )w({y∈Rn:ML(log L)1/ ( )(y)> }) + Csup >0 ϕ( )w({y∈Rn:Mα(T )(y)> }) whe e we also ha e used he doubling condi ion o ϕ. Nex , conside ing he es ima e M# α(T )(y)≤CαM (y),(3.21) which holds o all 0 < α < 1 (see [1], Theo em 2.1), i we u he selec αsuch ha 0 < δ < α < 1, he Fe e man-S ein’s lemma yields sup >0 ϕ( )w({y∈Rn:M# δ([b, T] )(y)> })≤ ≤Csup >0 ϕ( )w({y∈Rn:ML(log L)1/ ( )(y)> }) + Csup >0 ϕ( )w({y∈Rn:M# α(T )(y)> }) ≤Csup >0 ϕ( )w({y∈Rn:ML(log L)1/ ( )(y)> }) + Csup >0 ϕ( )w({y∈Rn:M( )(y)> }) ≤Csup >0 C ϕ( )w({y∈Rn:ML(log L)1/ ( )(y)> }), since i ially M( ) = ML( )≤ML(log L)1/ ( ). This inishes he p oo o (3.20). 20 3.4 The gene al case Suppose now ha (3.19) holds o m−1 and le ’s p o e i o m. Recall ha Φ( )=Φb( ) = log1/ (e+ ) wi h 1 =1 1+··+1 m. Then, by Lemma 3.1, sup >0 ϕ( )w({y∈Rn:M# δ(T~ b )(y)> }) ≤sup >0 ϕ( )w({y∈Rn:C MΦ( )(y) + m X j=1 X σ∈Cm j kbσkoscexpL σMε0(T~ bσ0 )(y) > }) ≤Cmsup >0 ϕ( )w({y∈Rn:MΦ( )(y)> }) +Cm m X j=1 X σ∈Cm j sup >0 ϕ( )w({y∈Rn:Mε0T~ bσ0kbσkoscexpL σ (y)> }) Since ε0<1 and we al eady checked ha he dis ibu ion se on he le hand side is ini e, combining he Fe e man–S ein’s lemma oge he wi h he induc ion hypo hesis on (3.19) o T~ bσ0we can es ima e he las exp ession by Cm m X j=1 X σ∈Cm j sup >0 ϕ( )w({y∈Rn:M# ε0T~ bσ0kbσkoscexpL σ (y)> }) ≤Cm m X j=1 X σ∈Cm j sup >0 ϕ( )w({y∈Rn:MΦ~ bσ0 (kbσ0koscexpL σ0kbσkoscexpL σ )(y)> }) ≤Cm m X j=1 X σ∈Cm j sup >0 ϕ( )w({y∈Rn:MΦ~ bσ0 ( )(y)> }) since k~ bσ0koscexpL σ0k~ bσkoscexpL σ=||~ b|| = 1. Finally, using he i ial obse a ion ha MΦ~ bσ0 ( )≤ MΦ~ bσ( ) = MΦ( ) we ha e sup >0 ϕ( )w({y∈Rn:M# δ(T~ b )(y)> })≤Cmsup >0 ϕ( )w({y∈Rn:MΦ( )(y)> }) and he claim (3.19) is p o ed. We need he ollowing lemma conce ning es ima es o he maximal ope a o MΦ, which is a mo e gene al e sion han he one gi en in Lemma 8.3 [20]. The p oo is s anda d and we shall omi i . Lemma 3.2 Le w∈A1, hen he e exis s a posi i e cons an Csuch ha o any > 0and any locally in eg able unc ion w({y∈Rn:MΦ (y)> })≤CZRn Φ(| (y)| )w(y)dy. 21 We a e now in posi ion o p o e Theo em 1.5 P oo o he Theo em 1.5: By homogenei y i is enough o assume =||b|| = 1 and hence we mus p o e w({y∈Rn:|Tb (y)|>1})≤CZRn Φ(| (y)|)w(y)dy. Now, since Φ is submul iplica i e, namely Φ(ab)≤2Φ(a) Φ(b), a, b ≥0 we ha e by Theo em 1.6 and Lemma 3.2 w({y∈Rn:|Tb (y)|>1})≤Csup >0 1 Φ(1 )w({y∈Rn:|Tb (y)|> }) ≤Csup >0 1 Φ(1 )w({y∈Rn:MΦ (y)> }) ≤Csup >0 1 Φ(1 )ZRn Φ(| (y)| )w(y)dy ≤Csup >0 1 Φ(1 )ZRn Φ(| (y)|)Φ(1 )w(y)dy ≤CZRn Φ(| (y)|)w(y)dy and he p oo is concluded. Re e ences [1] Al a ez, J., and P´ e ez, C.,Es ima es wi h A∞weigh s o a ious singula in eg al ope a o s, Boll. Un. Ma . I al. A (7) 8(1994), no. 1, 123–133. 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Ca los P´ e ez Depa men o de An´ alisis Ma em´ a ico Facul ad de Ma em´ a icas Uni e sidad de Se illa 41080 Se illa Spain E-mail add ess: [email p o ec ed] Rod igo T ujillo-Gonz´ alez Depa men o de An´ alisis Ma em´ a ico Uni e sidad de La Laguna 38271 La Laguna - S/C de Tene i e Spain E-mail add ess: [email p o ec ed] 24